Strategy for Finding the Best Fit Equation of a Parabola

We are looking for a rule of the form , where h is the x value of the vertex and k is the y value of the vertex. Since the vertex in our example appears to be at (2,5), we can fill in this information as follows:

The value of a is a little more involved. First, since the parabola is opening down, we know that there has been a vertical flip and a must be a negative number. Below is a graph comparing the parabola we are given and our evolving rule with a substitution of a = -1.


Notice we have chosen . We have not yet taken into account any vertical stretch or compression. Finally, since the graph we want (purple) appears to be "skinnier" than our evolving graph (red), we need to stretch our graph vertically (make it appear "taller" or "skinnier"). In order to do this, we know that . From here we can use a graphing utility and guess and check to get our best fit graph.